I enjoy every aspect of the quilting process - but I don't do all cutting, followed by all sewing etc. I cut some, sew some, cut some, join a row or two....you get the idea.
The most enjoyable part (for me) however is hand-stitching the binding down at the end. By the time I get to that part I've had a fairly long, love-hate relationship with the project, and in this final "act" I realize just how much I love it after all :D